Radius of Convergence (Ratio Test) – AP Calculus BC

Learning Objectives

Apply the Ratio Test to find the radius of convergence $R$.

Identify open/closed boundaries of the interval.

Visually connect infinite Taylor Polynomials exploding outside $|x-c| < R$.

Ratio Test

$ \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 $

Resolves to: $|x - c| < R$
where $R$ is the Radius of Convergence.

Why It Matters

A Taylor Series is an infinite polynomial approximation of a mathematical function. However, the approximation only holds true (converges) within a strict radius. Outside this golden boundary, the math literally explodes to infinity.